Render only changing part in the animation
Lets say it depends on the level of detail and realism you want. Mostly it has to do with the lighting and shadows. If you don't mind if the shadows aren't animated, then you can easily render just the static objects on to a transparent background, as a simple image. Then you can render out the dynamic parts onto a transparent background, then use the compositor and put it all together as the animation. If you want the dynamic shadows then it can be done as render layers, but the shadows will effect the chessboard and require it to be rendered. UV mapping and baking the materials will help with the render time. you will just want the diffuse color, none of the lighting or other effects. That will only help if the materials and textures are complex.